Abstract: Problem B: Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.

Algebra Etc. Seminar: Thursday, April 29, 4:00pm in Olin 383

"Isoperimetry in SL(n,Z)", Stephen Wang (Bucknell University)

Abstract: The isoperimetric function, or Dehn function, of a space measures how much area is enclosable with a loop of length at most n. Translated to a group theory context, this asks how quickly a word of length n in the generators that represents the identity can be shown to do so by application of the defining relations. I will discuss the relationship between the algebraic question and the geometric question, particularly with regard to the groups SL(n,Z), for which the Dehn function is now known except in the case n=4, and sketch some approaches to the problem.

Abstract: A modular form is basically a function on the complex upper half plane that satisfies a nice transformation property. This allow us to also think about a modular form as a Fourier series sum a(n)q^n, where q:= exp(2 pi i z). This transformation property might seem a bit loose, but actually imposes some interesting structure to the set of modular forms. I am currently interested in two particular properties. One is that the zeros of certain forms actually restrict themselves to a very specific location. The other is that there is a strange duality between the Fourier coefficients of certain sequences of modular forms. I will be talking about these two properties and their relation.

Distinguished Visiting Professor, April 19-24

Nathan Feldman (Washington & Lee University)

Hosted by Paul McGuire. Professor Feldman will give a faculty colloquium and an Analysis seminar.

Abstract: We will give a survey of the many different types of orbits that a linear operator on a Hilbert space may have, including some classic results, some very recent results, and some unsolved problems.

Abstract: We will discuss more carefully the new classes of n-weakly hypercyclic operators and n-weakly supercyclic operators and discuss some examples and open questions surrounding these ideas. Surprisingly, these new ideas are interesting and nontrivial even for 2x2 matrices!

Distinguished Visiting Professor, April 9-16

James Wilson (The Ohio State University)

Hosted by Peter Brooksbank. Professor Wilson will be the speaker of the student colloquium this week; he will also give a faculty colloquium.

Student Colloquium: Thursday, April 15, 12:00 noon in Olin 268

"Counting to Octonions: the final numbers and why they matter", James Wilson (The Ohio State University)

Abstract: Algebra and Analysis are two of the pillars in mathematics. Algebra tries to solves equations and so often studies numbers α and β where: • there is a unique x such that α + x = β, and • if α = 0, then there is a unique x such that αx = β. Analysis measures things and so looks at numbers α with a magnitude |α| along with the following rules which make pictures possible: • |α|≥ 0 and |α| = 0 if, and only if, α = 0; • for all numbers α and β, |αβ| = |α|•|β|; and • for all numbers α and β, |α + β|≤|α| + |β|. The best numbers would need to work for Algebra and Analysis. So, what numbers satisfy all these rules? The answer is the composition algebras which include the well-known real numbers R and the lesser known complex numbers C.A triumph in the 1800’s was the discovery of last two: Hamilton’s quaternions H, and Graves’ octonions O. Though the quaternions and octonions can be bizarre, they have remarkable uses in graphics, electromagnetism, quantum mechanics, group theory, and Lie theory – which is a math based on the product-rule from the calculus. This talk will develop these numbers and show how they appear in so many places. Most of the talk should be accessible to those who remember high-school algebra.

Faculty Colloquium: Thursday, April 15, 4:00pm in Olin 372

"Neo-classical geometry, division, and adjoints", James Wilson (The Ohio State University)

Abstract: Classical geometry studies non-degenerate forms on ﬁnite-dimensional vector spaces and leads to the creation of many of the simple objects in algebra. A neo-classical geometry can be described as the intersection of various classical geometries. Unlike the classical case, the groups and algebras associated to neo-classical geometries have a structure very far from simple objects. We associate to each neo-classical geometry a self-dual Grothendieck category of adjoints. We develop products, radicals, semisimple objects, and extensions in a manner similar to rings and modules. The algebraic invariants are functorial. Yet, this promising start encounters two surprises which greatly distinguishes these category from other popular categories. First there are no free objects, no projectives, no injectives, and so there are no presentations and no usual cohomology. Second, the simple objects cannot be parameterized through Morita equivalence. Indeed, the nondegenerate-simple objects in this category are exactly the division maps (so no analogue of matrices), and include all nonassociative division rings – a rich but often difficult family of objects.

Distinguished Visiting Professor, April 5-9

Eric Bahuaud (Stanford University)

Hosted by Emily Dryden. Professor Bahuaud will be the speaker of the student colloquium this week, he will also give a faculty colloquium.

Abstract: In 1904, while working on the foundations of topology, Poincaré asked whether or not every closed three-dimensional manifold with the property that every closed loop can be continuously shrunk to a point was homeomorphic to the three sphere. That this should be true became known as the Poincaré conjecture and defied attempts at proof for over 99 years. The Clay Mathematics Institute even announced that a resolution of the conjecture would carry a prize of $1,000,000. In 2003, building on the work of Hamilton and others, Perelman used the Ricci flow -- a system of differential equations that 'evolve geometry' -- to finally prove the conjecture. In exciting recent news, earlier this month the Clay Institute confirmed that Perelman had met the requirements to be awarded the $1,000,000.

Clearly the Ricci flow has played an important role in mathematics. However even before 2003 the Ricci flow was a useful tool in geometric analysis, and its importance continues to grow. In my talk I will give an introduction to the Ricci flow. I will begin with an overview of curvature of a Riemannian manifold and then review the model differential equation. I'll finish by discussing some examples and properties of the flow.

Abstract: The Fourier transform is a remarkable tool from mathematical analysis. We can think of it as a function on functions and it has the property that it can change a differential equation into an algebraic one. In this talk I will introduce the transform, outline some of its properties and show you how to convert calculus to algebra to solve an important PDE from physics.

Algebra Etc. Seminar: Thursday, April 8, 4:00pm in Olin 383

"A Pieri Rule for Skew Shapes", Peter McNamara (Bucknell University)

Abstract: We say that a polynomial is symmetric if it is invariant under any permutation of its variables x1, x2, ... , xn. The symmetric polynomials that are homogeneous of degree n form an algebra over the rational numbers. For reasons we will discuss, the Schur polynomials are often considered to be the most important basis for this algebra, and much attention has been given over the years to finding simple expressions for the product of two Schur polynomials. The Pieri rule, which dates to 1893, gives a beautiful such expression in an important special case. Last summer, Sami Assaf and I stumbled across an extension of the Pieri rule to skew Schur polynomials, which are perhaps the most well-known generalization of Schur polynomials. Surprisingly, it appears that our extension is new. I will elaborate on all of the above, as well as some ongoing work.

To all Bucknell students. Swing by for an informal gathering of Pizza π's, Fruit π's, Salad π's, Chocholate π's to (belatedly) celebrate that most delicious ratio. This event is presented by the MAA Club-Bucknell University Students.

Abstract: The first part of this talk will be expository: for a particular elliptic curve E, I will describe the group of points of E over different fields, the definition of its L-series, Birch and Swinnerton-Dyer conjecture, quadratic twists, and modularity.

In particular, we will recall a result of Kohnen and Waldspurger which relates central values of L-series of quadratic twists of E with coefficients of a particular modular form of weight 3/2.

In the second part I will show how the theory of p-adic families of modular forms enables us to give results about central /derivatives/ of the L-series of quadratic twists of E. These results are joint work with Henri Darmon.

Abstract: As proposed by Birch, one can construct partial Brandt matrices by the method of neighboring lattices for ternary quadratic forms.

In this talk we will present a refinement of the classical notion of proper equivalence of lattices which leads to the construction of the full Brandt matrices, at least in the squarefree level case. Moreover this refinement leads naturally to (and is motivated by!) the definition of generalized ternary theta series.

We apply these ideas to the construction of modular forms of half integral weight, giving an explicit version of the Shimura correspondence which generalizes results of Eichler, Gross, Ponomarev, Birch, Schulze-Pillot, and Lehman.

Student Colloquium: Thursday, March 11, 12:00 noon in Olin 268

“Variety's the very spice of life, that gives it all its flavor,” Arthur Berg (Division of Biostatistics, Pennsylvania State University, Hershey)

Abstract: The variety of life is explored from a micro to macro scale--from genetics to diseases--illustrating how unexplained variability can be explained and how biostatistics can accelerate the realization of personalized medicine.

Abstract: Let Fn denote the free group of rank n generated by x1,...,xn and let G = (g1,...,gn) be an n-generator group. It is easy to see that there exists a group homomorphism θ: Fn → G given by θ(xi) = gi for i =1,...,n and that θ is onto. Thus, on setting R = ker(θ), we obtain a description of G as a factor group Fn/R. However, since G may well have many diﬀerent generating sets consisting of n elements, there will in general be many normal subgroups R of Fn such that G =Fn/R. We discuss a number of diﬀerent ways to 'compare' such normal subgroups and therefore diﬀerent ways of describing G.

Abstract: Let G be a d-generator group where d ≥ 2. In general, for each integer n ≥ d, there exist many n-element (ordered) generating sets for G. One of the most natural way to classify these generating sets is by collecting them into the Tn-systems of G. We discuss some applications of Algebraic K1-theory to the theory of Tn-systems of polycyclic groups.

Abstract: Elementary ideas can often inspire very deep thinking. We will consider one example of this, the sequence of partition numbers, which are based simply on whole number addition. Don't be fooled, though, by the simplicity of arithmetic. We will see how intriguing questions arise while studying this sequence, leading us to a variety of powerful mathematical techniques. We will also think about how small changes to this sequence can lead to sum fun results.

Faculty Colloquium: Thursday, February 25, 4:00pm in Olin 372

“Making sense of mathematics: What can we learn from the Farmer Brown problem?,” Kay Wohlhuter (Department of Education, University of Minnesota, Duluth)

Abstract: The Farmer Brown problem is accessible to learners of all ages. Preservice teachers’ examination of solutions by kindergarteners through tenth-graders challenged them to look beyond the answer in order to make sense of students’ mathematical thinking. In the process, teachers recognized students’ sophisticated thinking as well as their misconceptions.

Algebra Etc. Seminar: Thursday, February 18, 4:00pm in Olin 383

"What's modularity and what am I doing about it?", Nathan Ryan (Bucknell University)

Abstract: When Taylor and Wiles proved Fermat's Last Theorem, they proved an instance of modularity. Modularity conjectures prescribe a way to explicitly connect arithmetic objects (elliptic curves, Galois representations, abelian varieties) to analytic ones (modular forms). In this talk I will try to give you some idea of how this can be done and to describe ongoing work of mine (with Gonzalo Tornaría) in this area.

Abstract: I will present an introduction to modeling and simulation focusing on a few different real-world problems, which use simulation for evaluating systemprototypes. The talk will introduce the general workflow of the M&S process and discuss some of the current challenges in the field.

Abstract: Given a representative for a homology class it is often difficult to determine whether or not the homology class is zero. We will discuss a specific computation of this type that arose in the study of subspace arrangements. This talk will focus on the algebraic aspects of the computation using linear algebra and the combinatorics of hypergraphs.

Abstract: A quintessential problem in the theory of symmetric functions is to prove that a given function is symmetric and Schur positive. Better still, one hopes that such a proof will provide a combinatorial interpretation for the Schur coefficients. For example, if the function arises as the (possibly graded) character of a representation of the symmetric group or general linear group, then the Schur expansion gives the irreducible decomposition of the representation. In this talk, we present a general framework for giving a combinatorial proof of symmetry and Schur positivity of an arbitrary function expressed as a sum of quasi-symmetric functions. This method uses a combinatorial construction called dual equivalence graphs (or, more generally, D graphs), which in some ways act as the symmetric group analog of crystal graphs for the general linear group. We will outline the construction of D graphs for several classes of functions, including Macdonald polynomials, and point to some connections with representation theory.

Abstract: LLT polynomials are q-analogs of products of Schur functions defined by Lascoux, Leclerc and Thibon. In this talk we will discuss several aspects of these polynomials, including a combinatorial proof of Schur positivity, and show how they arise in many guises in connection with Macdonald polynomials.

Abstract: The Lambert function W(x) (sometimes called the “Omega” function) is easily defined and available in the software packages Maple and Mathematica. Despite this, its properties, including those related to its derivative, usually seem to be overlooked. Following a brief introduction of this function and its derivative, examples of the application of the related function W(a Exp[a-bt]) in chemical kinetics will be presented.

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